Research Division Federal Reserve Bank of St. Louis Working Paper Series

Identifying Technology Shocks in the Frequency Domain

Riccardo DiCecio and

Michael T. Owyang

Working Paper 2010-025A http://research.stlouisfed.org/wp/2010/2010-025.pdf

August 2010

FEDERAL RESERVE BANK OF ST. LOUIS Research Division

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Identifying Technology Shocks

in the Frequency Domain1

Riccardo DiCecio

Federal Reserve Bank of St. Louis

Michael T. Owyang

Federal Reserve Bank of St. Louis

First Draft: June 2010

This Draft: August 5, 2010

1We are grateful to N. Francis, T. Lubik, V. Ramey, and H. Uhlig for comments and discussions.

Kristie M. Engemann provided research assistance. All errors are our own. Any views expressed

are our own and do not necessarily re�ect the views of the Federal Reserve Bank of St. Louis or

the Federal Reserve System. Corresponding author: Riccardo DiCecio, dicecio@stls.frb.org.

Abstract

Since Galí [1999], long-run restricted VARs have become the standard for identifying the

e¤ects of technology shocks. In a recent paper, Francis et al. [2008] proposed an alterna-

tive to identify technology as the shock that maximizes the forecast-error variance share of

labor productivity at long horizons. In this paper, we propose a variant of the Max Share

identi�cation, which focuses on maximizing the variance share of labor productivity in the

frequency domain. We consider the responses to technology shocks identi�ed from various

frequency bands. Two distinct technology shocks emerge. An expansionary shock increases

productivity, output, and hours at business-cycle frequencies. The technology shock that

maximizes productivity in the medium and long runs instead has clear contractionary e¤ects

on hours, while increasing output and productivity.

JEL: C32, C50, E32.

Keywords: Aggregate �uctuations; Business cycle; Frequency domain; Technology shocks;

Vector autoregressions.

1 Introduction

Estimated impulse responses identi�ed from VARs are often used to distinguish between

competing macroeconomic theories. For example, using long-run restrictions, Galí [1999]

and Francis and Ramey [2005] found that hours decline in response to a positive technology

shock and argued that this provided evidence invalidating the standard RBC model in favor

of sticky price models. This conclusion, however, appears to depend on the speci�cation

of hours in the VAR.1 The salient result is that if speci�ed in di¤erences, hours contract.

On the other hand, Christiano et al. [2003] found that if speci�ed in levels, hours respond

positively on impact. Arguing that hours per capita should be stationary fails to provide

any resolution: Pesavento and Rossi [2005] showed that when the hours process is presumed

stationary but close to a unit root, hours contract.

More recently, a few papers have attempted to resolve these con�icting results by exam-

ining the manner in which productivity and hours are detrended. Fernald [2007] proposed

that productivity growth and hours had simultaneous shifts in mean. He argued that the

shifts in productivity growth did not re�ect technological innovation and should, therefore,

be removed. One solution was to demean the series, while allowing for multiple breaks in

the intercept, which con�rmed the �nding of a contraction in hours.2 Francis and Ramey

[2009] argued that the apparent unit root in hours was caused by changing demographic

trends; once properly detrended, hours declined on impact.3 Gospodinov et al. [2009], on

the other hand, argued that if one is interested in preserving the underlying comovements

between productivity and hours, the VAR should be estimated with hours in levels and not

detrended. In this case, they found that hours rise following a technology shock. Thus,

1Others have used sign restrictions to identify technology shocks. Dedola and Neri [2007] employed

restrictions recovered from a DSGE model. Peersman and Straub [2007] used restrictions recovered from a

New Keynesian model. Both found neutral technology shocks to be expansionary on impact.2This is similar to the recommendation of Canova et al. [2010], who argue against detrending the hours

series.3Galí and Rabanal [2005] also showed that neutral technology shocks are contractionary when hours is

detrended.

1

how the VAR distinguishes between models appears to depend on whether one believes the

low-frequency comovements can be attributed to technology.

Although long-run restrictions have been the popular choice for identi�cation, macro

models also have implications for the e¤ect of technology shocks at non-zero frequencies.

For example, the RBC literature [see Prescott, 1986] typically assumes that the business-

cyle frequencies of the data (hours, output, investment, consumption) are driven solely by

technology shocks.4 Thus, an alternative to identifying technology shocks using long-run

restrictions is to impose restrictions on the e¤ect of technology shocks at non-zero frequencies.

In particular, we identify the shock that maximizes the share of the forecast-error variance

(FEV) in productivity growth and hours at various frequencies in a standard bivariate VAR.

We consider business-cycle frequencies identi�ed using linear �lters: the HP1600 �lter and

the band-pass (BP) �lter for periods between 8 and 32 quarters. We also consider medium-

term cycles identi�ed by the BP �lter for periods of either 32-80 or 80-200 quarters. Our

identi�cation is adapted to the frequency domain from the method introduced by Faust

[1998] and extended to technology shocks in Francis et al. [2008] (FOR).5 We analyze the

e¤ect of varying the frequency window on the identi�ed the shocks. As in Fernald [2007],

we account for the low-frequency comovement between productivity growth and hours by

demeaning the two series and allowing for two breaks in mean.

Two distinct technology shocks emerge. When identi�ed from frequencies lower than

the business cycle, technology shocks have Schumpeterian e¤ects [Caballero and Hammour,

1994, 1996, Canova et al., 2007]. While productivity increases persistently, output responds

ambiguously on impact, with both positive and negative draws, and increases eventually.

These shocks have a clear contractionary e¤ect on hours, which fall on impact and converge to

4Hansen [1997] pointed out that RBC models driven by persistent, but stationary, technology shocks and

models driven by I(1) shocks have similar implications at business-cycle frequencies.5The identi�cation in FOR is similar in �avor to Uhlig�s [2004] medium-run identi�cation. Uhlig argued

that other shocks (e.g., dividend tax shocks) could have long-run e¤ects on productivity and, thus, confound

the Galí [1999] identi�cation. Like FOR, Uhlig computed the FEV share but, instead, calibrated it to a

value determined by Monte Carlo simulations from a theoretical model.

2

zero from below. Interestingly, the shocks that drive productivity in the medium term (32-80

and 80-200 quarters) are indistinguishable from each other and from the shock identi�ed with

long-run restrictions. On the other hand, when identi�ed from business-cycle frequencies,

technology shocks are expansionary, inducing positive comovement of ouptut, hours, and

productivity, and are broadly consistent with the predictions of an RBC model. The shocks

identi�ed by the HP1600 and the BP8;32 �lters are almost identical.

The balance of the paper is organized as follows: Section 2 describes the baseline struc-

tural VAR and the identi�cation problem. In this section, we also discuss the issues of

inference using long-run restrictions in small samples. Section 3 presents our alternative

frequency-based identi�cation. Section 4 compares the results across di¤erent frequency

windows used for identi�cation. Section 5 summarizes and o¤ers some conclusions.

2 The Baseline Structural VAR

Consider the reduced-form VAR(p) representation of the (n� 1) vector of variables Yt:

Yt = A (L)Yt�1 + ut; (1)

where A (L) is a matrix polynomial in the lag operator, L, of order p. The reduced-form

residuals, ut, have zero mean, i.e., E (ut) = 0, and covariance matrix �, i.e., E (utu0t) = �.

We can rewrite (1) as

Yt = A (L)Yt�1 + C"t;

where C � (A0)�1. The structural matrix, A0, maps the reduced-form residuals into the

structural shocks, "t, where E ("t"0t) = I.

The central issue in the structural VAR literature involves the identi�cation of the struc-

tural matrix, A0. It is well known that in�nitely many possible structural matrices can be

derived from the decomposition of the reduced-form variance-covariance matrix, �. Various

methods have been used to impose su¢ cient identifying restrictions. Short-run identi�ca-

3

tions, for example, impose restrictions directly on the elements of the structural matrix.

While short-run identi�cation schemes have proven less controversial econometrically, they

typically are imposed through (potentially ad hoc) recursive ordering restrictions and may,

therefore, be less economically appealing.

Long-run restrictions, on the other hand, are often more consistent with theoretical mod-

els. For example, technology shocks are often thought of as important drivers of business

cycles in RBC models. Thus, the empirical literature has used VARs to identify technology

shocks assuming that they are the sole source of the unit root in labor productivity [see,

e.g., Galí, 1999]. This identifying assumption is implemented in VARs by imposing that non-

technology shocks have no long-run� i.e., the frequency-zero� e¤ect on labor productivity.

Long-run restrictions identify the structural matrix, A0, using restrictions of the form

�[I � A (1)]�1C

ij= 0; (2)

which impose that shock j has no long-run e¤ect on variable i.

Long-run restrictions� despite their theoretical appeal� have been shown to have some

drawbacks. Based on arguments initially made by Sims [1972] and Faust and Leeper [1997],

Christiano et al. [2006] contended that, in general, biases in the impulse responses of a

VAR depend on two components: propagation (i.e., misspecifying the companion matrix

for the VAR) and identi�cation of the structural matrix (i.e., incorrectly choosing C). The

former should be invariant to the identi�cation strategy. Christiano et al. [2006] advocated

for short-run restrictions on the basis that identi�cation of C relies on the estimate of �

only. Identi�cation of C via long-run restrictions, on the other hand, relies on the estimate

of � and the estimates of the As via the use of A (1) in (2). Because the VAR is truncated,

the estimate of A (1) may have greater bias when the actual share of the spectral density

of the data is not large near frequency zero. That is, we may encounter inference problems

when we use the spectral density at the zero frequency as an input to the identi�cation if

there is not much information at the zero frequency.

In many models, there is a long-run interpretation of the shocks, and the spectral density

4

near frequency zero may be su¢ ciently high to accurately estimate A (1). But what happens

when it is not? The solution proposed by Christiano et al. [2006] is an alternative estimator,

based on the Bartlett estimator which is aimed at estimating A (1) more accurately. Another

alternative advocated by FOR is to examine theoretically-consistent identifying restrictions

that do not depend so heavily on the zero frequency.

3 Identifying Shocks in the Frequency Domain

We describe how to identify the structural shock that maximizes (or minimizes) the share

of the FEV of a given variable in the frequency domain.6 The method can be applied to

any frequency window, including the entire spectrum. De�ne the matrix D as the Cholesky

factor of � and & i as the ith colum vector of D. We will refer to the set of & i as the Cholesky

shocks. Then, all of the possible impulse vectors, � (�), resulting from exact identi�cation

can be expressed as a linear combination of the Cholesky shocks as follows: � (�) = �0D,

where (�0�) = 1. Put di¤erently, � is a column of a rotation matrix that maps the Cholesky

shocks into another set of structural shocks. The goal of the procedure that follows is to

generate �.

In particular, we consider the component of the FEV of the variable of interest identi�ed

by a linear �lter, � (!). For convenience, we can �rst map the variables in Yt into other

variables of interest in the vectors ~Yt and �Yt as follows:

�Yt = F (L)�1 Yt; (3)

~Yt = G (L) �Yt = G (L)F (L)�1 Yt; (4)

where F (L) and G (L) are matrix polynomials in the lag operator. The matrix F (L) may

be used, for example, to convert some of the elements of Yt into di¤erences, while G (L)

rede�nes some variables (e.g., transforming output and hours into labor productivity).

6The construction of the FEV shares in the frequency domain follows Altig et al. [2005].

5

The spectral density of ~Yt is given by:

S ~Y�e�i!

�= H

�e�i!

��H

�e�i!

�0; (5)

where H (e�i!) �nG (e�i!)F (e�i!)

�1[I � A (e�i!)]�1

o. Given a structural shock (i.e., a

vector �) and a linear �lter with gain � (!), the share of the FEV of variable j due to � is

given by7:

V ~Yj (�) = �0

hR ��� � (!)S �Yj (e

�i!) d!i

trhR ��� � (!)S �Yj (e

�i!) d!i�: (6)

The shock with the largest share of the FEV of variable j associated with the �lter � (!)

is given by �� = argmax� V ~Yj (�). In principle, the choice of the �lter, � (!), can be ad

hoc (e.g., HP �lter) or governed by theory. For example, the standard long-run restriction

would amount to maximizing the FEV as ! approaches zero. In practice, we can utilize a

set of linear �lters constructed such that the gain re�ects, say, business-cycle or medium-run

frequencies.

4 Technology Shocks

In this section, we analyze the e¤ects of technology shocks in a bivariate VAR in labor

productivity growth,� log (Y=H), and hours per capita, log (H=N).8 We consider technology

shocks identi�ed by the algorithm proposed in Section 3 for a variety of frequency windows.

The sample covers the period 1948:Q2-2009:Q4. Previous studies have highlighted changes

in the mean growth of productivity over the sample. We follow Fernald [2007] and demean

7The integrals in equation (6) can be computed by Gauss-Legendre quadrature [see Press et al., 2007, p.

183].8Productivity growth is measured as the annualized log-di¤erence of output per hour of all persons in the

nonfarm business sector; hours per capita are included in logs and measured as hours of all persons in the

nonfarm business sector scaled by the civilian noninstitutional population (16 years and over). The Haver

mnemonics in the USECON database for productivity, hours, and population are LXNFA, LXNFH, and

LN16N, respectively.

6

both productivity and hours, allowing for two breaks: 1973:Q2 and 1997:Q2 (see Table 1).

Figure 1 portrays the two time series and their corresponding sample spectral densities after

demeaning.

Using the notation introduced above, we have:

Yt =

�� log

�Y

H

�; log

�H

N

��0;

�Yt =

�log

�Y

H

�; log

�H

N

��0= F (L)�1 Yt;

~Yt =

�log

�Y

N

�; log

�Y

H

�; log

�H

N

��0= G (L) �Yt:

The matrix polynomials, F (L) and G (L), that allow to recover productivity, log (Y=H), and

output per capita, log (Y=N), are given by:

F (L) =

24 1� L 0

0 1

35 ;

G (L) =

266641 �1

1 0

0 1

37775 :The reduced-form VAR is estimated with four lags, i.e., p = 4.

To estimate the reduced form of the VAR, we utilize Bayesian techniques. For the Gibbs

sampler, we impose a Je¤reys [1961] prior on the reduced-form VAR parameters9:

p (A;�) / det (�)�n+12 ; (7)

where A = [A1; :::; Ap]0. The posterior distribution of the reduced-form VAR parameters

belongs to the inverse Wishart-Normal family:

(�jYt=1;:::;T ) � IW�T �; T � k

�; (8)

(Aj�; Yt=1;::;T ) � N�A;� (X 0X)

�1�; (9)

9We truncate the prior to rule out posterior draws that imply explosive roots in the VAR.

7

where A and � are the OLS estimates of A and �, T is the sample length, k = (np+ 1),

and X is de�ned as

X =hx01; :::; x

0

T

i0;

x0t =h1; Y 0t�1; :::; Y

0

t�p

i0:

We identify �ve di¤erent �Max Share� technology shocks that maximize the FEV of

productivity at di¤erent frequencies:

1. the business-cycle frequencies extracted by an HP �lter of parameter � = 1; 600;

2. the business-cycle frequencies with a period between 8 and 32 quarters;

3. the frequencies corresponding to �uctuations with a �medium-run�period between 32

and 80 quarters;

4. the frequencies corresponding to �uctuations with a �medium-run�period between 80

and 200 quarters; and

5. the entire set of frequencies in [��; �] :

The gain functions of the corresponding linear �lters are reported in Table 2. The fre-

quencies corresponding to a given interval of periods, [l; u], are extracted with a BP �lter

[Christiano and Fitzgerald, 2003]. For comparison, we also consider the long-run identi�ca-

tion strategy proposed by Galí [1999].

For each draw from the posterior distribution of the reduced-form VAR parameters given

by equations (8) and (9), we identify a technology shock and construct impulse response

functions (IRFs) and FEV shares. For each of the identi�cation schemes described above,

we obtain 5; 000 draws from the posterior distribution of IRFs to various technology shocks

and the corresponding FEV shares at various horizons.

8

4.1 Results

We plot the median, 16th, and 84th percentiles of the posterior distributions of the impulse

responses to a one-percent increase in productivity growth in Figures 2-3. As a basis for

comparison, the impulse responses to the technology shock identi�ed by the standard long-

run restrictions are shown in the last column of Figure 3.

Two distinct technology shocks emerge from our identi�cation schemes. The shock that

maximizes the FEV of productivity at business-cycle frequencies� BP8;32 and HP1600 produce

almost indistinguishable IRFs� resemble the shock that drives RBC models. It induces

positive comovement between productivity, output, and hours. The responses of output and

hours are persistent and hump-shaped.

When maximizing the FEV of productivity at lower frequencies, the same shock emerges

from di¤erent identi�cation schemes� BP32;80, BP80;200, the entire set of frequencies, and

long-run restrictions. On impact this shock generates an ambiguous output response with

both positive and negative draws; eventually, output increases. This shock has a clear

contractionary e¤ect on hours: The median IRF is negative on impact and converges to zero

from below. The persistent increase in productivity is similar to the one generated by the

business-cycle shock described above.

Recall that the identi�ed technology shock is a linear combination of the Cholesky shocks,

where ��� the appropriate column of a rotation matrix� gives the weights. Figure 4 shows

the responses to the two Cholesky shocks. Figure 5 shows the posterior distributions of the

rotation angle between the two Cholesky shocks, �� = atan2 (��2; ��1), where �

�1 and �

�2 are

the two elements of ��. First, note that, for the bivariate system, the �rst Cholesky shock

(shown in the �rst column of Figure 4) appears consistent with the shock taken from the

standard RBC model. This would suggest that, for the shock identi�ed by the HP1600 or

the BP8;32 �lter, the rotation angle between the two Cholesky shocks should be centered at

zero. The �rst panel of Figure 5 reveals this to be true. For all the other frequency windows

we consider, more weight is placed on (the negative of) the second Cholesky shock. The

negative of this shock is similar to a contractionary technology shock.

9

Tables 3-5 show how much of the FEV of each variable the shocks explain at various

horizons. The last row of Table 3 reports the Max Share of productivty associated with each

shock, Vlog(Y=H) (��). Both the business-cycle and low-frequency technology shocks explain

most of the FEV of productivity at all horizons. The low-frequency shock�s share approaches

100% as the horizon increases. In contrast, the business-cycle shock explains almost all of

productivity at short horizons and more than 60% at long horizons.

The business-cycle frequency shock explains more than 50% of the FEV of output at all

horizons. The medium-term/low-frequency shock explains very little of the FEV of output

at short horizons� less than 15% up to two years. The FEV share of this shock increases

above 70% at long horizons.

Both shocks we identify account for less than 25% of the FEV of hours at any horizon.

The �nding that technology shocks account for little of the variance of hours is common-

place in the literature. The FEV share of the business-cycle (low-frequency) shock increases

(decreases) with the forecast horizon.

5 Conclusions

Since Galí [1999], technology shocks have been identi�ed in VARs using long-run restrictions.

In a recent paper, FOR� building on Faust [1998]� proposed a �nite-horizon alternative to

the long-run identi�cation. Here, we extend their �nite-horizon alternative to the frequency

domain. We identify technology shocks by the rotation of the Cholesky factor that maximizes

the FEV for a chosen frequency window. The frequency domain provides natural identifying

restrictions as many macro models have implications for technology shocks at business-cycle

frequencies.

Using a bivariate VAR with productivity and hours, we compare the identi�cation over

a number of frequency windows. The resulting impulse responses are broadly consistent

for frequency windows outside the business cycle. The shock identi�ed by these frequencies

is contractionary in hours with essentially zero e¤ect on output on impact. On the other

10

hand, the shock identi�ed by maximizing the FEV for business-cycle frequencies is more

consistent with an RBC technology shock� hours rise weakly on impact and output rises

unambiguously.

Another fundamental di¤erence between the two types of shocks identi�ed by the di¤erent

frequency windows is at what horizon the shock has the greatest e¤ect. For the contrac-

tionary shock, technology accounts for a relatively small share of the short-horizon FEV of

productivity and output. At longer horizons, this shock explains more of the variation in

both those variables. While the shock never explains a large portion of the FEV in hours, the

amount explained declines as the forecast horizon grows. The shock identi�ed by business-

cycle frequencies� the shock most consistent with the RBC shock� behaves in the opposite

manner, explaining more of productivity and output at short horizons and accounting for

less variation as the horizon grows.

11

References

D. Altig, L. J. Christiano, M. Eichenbaum, and J. Linde. Technical appendix to: Firm-

speci�c capital, nominal rigidities and the business cycle. URL http://faculty.

wcas.northwestern.edu/~lchrist/research/ACEL/technical.pdf. unpublished man-

uscript, Northwestern University, July 2005.

R. J. Caballero and M. L. Hammour. The cleansing e¤ect of recessions. American Economic

Review, 84(5):1350�68, December 1994.

R. J. Caballero and M. L. Hammour. On the timing and e¢ ciency of creative destruction.

Quarterly Journal of Economics, 111(3):805�52, August 1996.

F. Canova, D. Lopez-Salido, and C. Michelacci. Schumpterian technology shocks. unpub-

lished manuscript, CREi, Barcelona, Spain, November 2007.

F. Canova, D. Lopez-Salido, and C. Michelacci. The e¤ects of technology shocks on hours

and output: a robustness analysis. Journal of Applied Econometrics, 25(5):755�73, August

2010.

L. J. Christiano and T. J. Fitzgerald. The band pass �lter. International Economic Review,

44(2):435�465, May 2003.

L. J. Christiano, M. Eichenbaum, and R. Vigfusson. What happens after a technology shock?

NBER Working Paper No. 9819, July 2003.

L. J. Christiano, M. Eichenbaum, and R. Vigfusson. Assessing structural VARs. NBER

Macroeconomics Annual, 21:1�71, 2006.

L. Dedola and S. Neri. What does a technology shock do? a var analysis with model-based

sign restrictions. Journal of Monetary Economics, 54(2):512�49, March 2007.

J. Faust. The robustness of identi�ed VAR conclusions about money. Carnegie-Rochester

Conference Series on Public Policy, 49:207�44, 1998.

12

J. Faust and E. M. Leeper. When do long-run identifying restrictions give reliable results?

Journal of Business and Economic Statistics, 15(3):345�53, 1997.

J. G. Fernald. Trend breaks, long-run restrictions, and contractionary technology improve-

ments. Journal of Monetary Economics, 54(8):2467�85, November 2007.

N. Francis and V. A. Ramey. Is the technology-driven real business cycle hypothesis dead?

shocks and aggregate �uctuations revisited. Journal of Monetary Economics, 52(8):1379�

99, November 2005.

N. Francis and V. A. Ramey. Measures of per capita hours and their implications for the

technology-hours debate. Journal of Money Credit and Banking, 41(6):1071�97, September

2009.

N. Francis, M. T. Owyang, and J. E. Roush. A �exible, �nite-horizon identi�cation of

technology shocks. Federal Reserve Bank of St. Louis Working Paper No. 2005-024E, July

2008.

J. Galí. Technology, employment, and the business cycle: Do technology shocks explain

aggregate �uctuations? American Economic Review, 89(1):249�71, March 1999.

J. Galí and P. Rabanal. Technology shocks and aggregate �uctuations: How well does

the RBC model �t postwar U.S. data? In M. Gertler and K. Rogo¤, editors, NBER

Macroeconomics Annual 2004, pages 225�88, Cambridge, MA, 2005. MIT Press.

N. Gospodinov, A. Maynard, and E. Pesavento. Sensitivity of impulse responses to small

low frequency co-movements: Reconciling the evidence on the e¤ects of technology shocks.

CIREQ Cahier 03-2009, March 2009.

G. D. Hansen. Technical progress and aggregate �uctuations. Journal of Economic Dynamics

and Control, 21(6):1005�23, June 1997.

H. Je¤reys. Theory of Probability. Oxford University Press, London, 3rd edition, 1961.

13

G. Peersman and R. Straub. Putting the new Keynesian model to a test. unpublished

manuscript, Ghent University, June 2007.

E. Pesavento and B. Rossi. Do technology shocks drive hours up or down? a little evidence

from an agnostic procedure. Macroeconomic Dynamics, 9(4):478�88, September 2005.

E. C. Prescott. Theory ahead of business cycle measurement. Federal Reserve Bank of

Minneapolis Quarterly Review, 10(4):9�22, Fall 1986.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes.

Cambridge University Press, New York, third edition, 2007.

C. Sims. Money, income, and causality. American Economic Review, 62(4):540�52, Septem-

ber 1972.

H. Uhlig. Do technology shocks lead to a fall in total hours worked? Journal of the European

Economic Association, 2(2-3):361�71, April/May 2004.

14

Sampe period

Variable1948:Q2-

1973:Q1

1973:Q2-

1997:Q1

1997:Q2-

2009:Q4

400�� log�YH

�2:78 1:33 2:94

log�HN

��7:54 �7:56 �7:53

Table 1: Average productivity growth and hours p.c.

Filter � (!)

HP�4�[1�cos(!)]2

1+4�[1�cos(!)]2

BPl;u 1[ 2�u ;2�l ]

All 1[��;�]

Table 2: Gains of various �lters

Max Share Long

Horizon HP1600 BP8;32 BP32;80 BP80;200 All run

1 99:49[97:60;99:95]

98:04[90:85;99:03]

63:91[48:92;76:97]

62:20[37:13;83:30]

63:25[27:45;90:84]

60:94[19:75;92:53]

4 96:78[94:65;98:12]

96:15[91:02;97:87]

71:19[58:71;81:11]

70:00[46:76;86:79]

70:19[36:80;92:12]

68:10[27:98;92:88]

8 82:09[72:25;89:74]

83:91[70:67;91:09]

83:22[76:35;88:70]

82:13[68:25;90:08]

80:28[58:71;89:71]

79:01[51:69;89:51]

20 71:40[50:10;88:06]

73:60[46:42;91:46]

90:56[86:37;93:47]

91:03[85:36;94:82]

89:28[78:56;93:81]

88:50[73:92;93:52]

40 67:97[36:66;91:02]

69:39[32:93;93:33]

92:89[85:25;95:72]

94:83[91:61;96:82]

94:20[89:13;96:68]

93:82[86:46;96:50]

80 65:73[26:73;92:65]

66:32[22:58;94:07]

94:07[79:63;97:28]

96:64[93:38;97:96]

96:94[94:39;98:27]

96:85[93:63;98:21]

200 63:56[20:61;93:83]

63:38[16:39;94:91]

94:28[74:09;98:53]

97:99[92:99;98:97]

98:66[97:16;99:26]

98:74[97:65;99:28]

Vlog(Y=H) (��) 77:27

[70:24;83:82]72:29

[64:35;80:57]93:63

[83:89;97:93]97:85

[90:88;99:73]98:70

[97:76;99:18]100[�]

Table 3: FEV share of productivity due to technology shocks.

15

Max Share Long

Horizon HP1600 BP8;32 BP32;80 BP80;200 All run

1 59:94[48:62;70:32]

57:08[36:96;76:67]

6:95[1:42;16:26]

7:27[0:76;22:92]

9:93[0:94;39:05]

12:00[1:13;44:95]

4 51:44[40:22;61:80]

49:53[30:28;67:62]

3:94[0:88;11:42]

4:98[0:91;17:59]

8:18[1:11;34:22]

10:51[1:38;40:26]

8 59:47[48:63;69:49]

57:63[39:39;73:63]

8:49[2:39;18:76]

8:68[2:28;24;69]

11:70[2:62;40:93]

13:55[2:76;45:40]

20 68:47[59:29;76:19]

67:06[50:77;78:59]

19:76[10:88;30:39]

19:92[9:00;37:01]

22:62[8:71;51:64]

23:29[9:09;54:44]

40 71:08[62:36;77:72]

69:25[58:45;77:09]

35:37[23:82;45:38]

36:38[22:82;50:75]

39:35[22:30;62:07]

39:22[22:60;64:55]

80 70:70[53:55;80:23]

68:51[51:56;78:13]

52:57[39:69;62:13]

55:94[42:56;66:94]

58:99[43:80;74:47]

58:89[44:08;75:68]

200 67:37[35:78;86:03]

66:15[33:03;84:30]

71:31[53:73;79:70]

76:51[65:92;83:39]

79:39[69:94;87:60]

79:57[70:07;88:28]

Table 4: FEV share of output per capita due to technology shocks.

Max Share Long

Horizon HP1600 BP8;32 BP32;80 BP80;200 All run

1 2:34[0:29;7:18]

2:81[0:27;12:69]

20:67[9:52;35:22]

21:74[6:10;46:54]

22:59[3:32;57:14]

24:43[2:87;66:19]

4 13:14[5:40;22:92]

11:99[2:76;28:72]

7:72[2:33;18:41]

9:14[1:91;27:96]

11:97[2:06;40:86]

14:22[2:25;49:47]

8 20:58[10:08;32:31]

19:25[5:89;37:73]

4:73[1:94;13:19]

6:26[1:97;21:48]

9:75[2:31;35:38]

12:00[2:49;43:75]

20 22:70[11:32;34:95]

21:28[6:76;40:10]

4:22[1:57;11:89]

5:83[1:72;20:47]

9:27[2:23;34:35]

11:52[2:37;42:71]

40 22:99[11:42;35:37]

21:54[6:86;40:44]

4:12[1:46;11:83]

5:75[1:61;20:38]

9:23[2:18;34:28]

11:46[2:32;42:52]

80 23:09[11:42;35:45]

21:59[6:88;40:49]

4:11[1:43;11:84]

5:71[1:59;20:36]

9:25[2:17;34:27]

11:46[2:31;42:54]

200 23:10[11:42;35:50]

21:61[6:88;40:40]

4:11[1:42;11:86]

5:71[1:59;20:36]

9:24[2:17;34:23]

11:46[2:31;42:54]

Table 5: FEV share of hours per capita due to technology shocks.

16

1950 1960 1970 1980 1990 2000­0.02

­0.01

0

0.01

0.02

0.03

0.04Productivity growth

0 1 2 34

6

8

10

12x 10 ­5

1950 1960 1970 1980 1990 2000­7.7

­7.65

­7.6

­7.55

­7.5

­7.45

­7.4Hours p.c.

0 1 2 30

0.005

0.01

0.015

0.02

Figure 1: Data: time series (�rst row) and sample spectral densities (second row).

17

10 20 30 40

­0.5

0

0.5

1

1.5

Out

put p

.c.

Max Share ­ HP1600

10 20 30 40­1

­0.5

0

0.5

1

Hou

rs p

.c.

10 20 30 400

0.5

1

Prod

uctiv

ity

10 20 30 40

­0.5

0

0.5

1

1.5

Max Share ­ BP8,32

10 20 30 40­1

­0.5

0

0.5

1

10 20 30 400

0.5

1

10 20 30 40

­0.5

0

0.5

1

1.5

Max Share ­ BP32,80

10 20 30 40­1

­0.5

0

0.5

1

10 20 30 400

0.5

1

Figure 2: IRFs of hours, output, and productivity to various technology shocks (HP1600,

BP8;32, BP32;80).

18

10 20 30 40

­0.5

0

0.5

1

1.5

Out

put p

.c.

Max Share ­ BP80,200

10 20 30 40­1

­0.5

0

0.5

1

Hou

rs p

.c.

10 20 30 400

0.5

1

Prod

uctiv

ity

10 20 30 40

­0.5

0

0.5

1

1.5

Max Share ­ All freq.

10 20 30 40­1

­0.5

0

0.5

1

10 20 30 400

0.5

1

10 20 30 40

­0.5

0

0.5

1

1.5

Long run

10 20 30 40­1

­0.5

0

0.5

1

10 20 30 400

0.5

1

Figure 3: IRFs of hours, output, and productivity to various technology shocks (BP80;200, all

frequencies, long run).

19

5 10 15 20 25 30 35 40­0.5

0

0.5

1

1.5

2

Out

put p

.c.

Cholesky shock 1

5 10 15 20 25 30 35 40­1

0

1

Hou

rs p

.c.

5 10 15 20 25 30 35 40

­0.5

0

0.5

1

Prod

uctiv

ity

5 10 15 20 25 30 35 40­0.5

0

0.5

1

1.5

2

Out

put p

.c.

Cholesky shock 2

5 10 15 20 25 30 35 40­1

0

1

Hou

rs p

.c.

5 10 15 20 25 30 35 40

­0.5

0

0.5

1

Prod

uctiv

ity

Figure 4: IRFs of hours, output, and productivity to Cholesky shocks.

20

­3 ­2 ­1 0 1 2 30

1

2

3

4Max Share ­ HP

1600Max Share ­ BP

8,32Max Share ­ BP

32,80

­3 ­2 ­1 0 1 2 30

0.5

1

1.5

2Max Share ­ BP

80,200Max Share ­ All freq.Long Run

Figure 5: Posterior distributions of the rotation angle between Cholesky shocks, �� =

atan2 (��2; ��1).

21